ADVANCED DYNAMICS — PHY 4241/5227

HOME AND CLASS WORK – SET 14

(April 13, 2009)

(58) The electromagnetic field tensor transforms according to

F^{′ αβ} = a_{γ}^{α}a_{δ}^{β}F^{γδ} .

1. Choose the particular case of a Lorentz boost in x^{1}-direction, v⃗ = v ê_{1}, and write down the transformation law for the electric field E⃗ and the magnetic induction B⃗. Due April 15 before class (8 points).

2. Compare with the results you get from (due April 15 before class (4 points).

E⃗′ = γ (E⃗ + β⃗ × B⃗) − \frac{γ^{2}}{γ + 1}β⃗ (β⃗ E⃗) ,
B⃗′ = γ (B⃗ − β⃗ × E⃗) − \frac{γ^{2}}{γ + 1}β⃗ (β⃗ B⃗) .

(59) In the same way that the contraction, or relativistic dot product, of two four vectors is a Lorentz invariant, the contraction of two relativistic tensors is also a Lorentz invariant (Griffiths Problem 12.50).

1. Compute the three Lorentz invariants from the contraction of the tensors:

F^{μν}F_{μν} , ^{∗}F^{μν} ^{∗}F_{μν} and F^{μν} ^{∗}F_{μν}

in terms of the electric and magnetic fields E⃗ and B⃗. Due April 15 in class (3 points).

2. Suppose that in one inertial frame B⃗ = 0 but E⃗ ≠ 0 (at some point P). Is it possible to find another system in which the electric field is zero at P? Due April 15 in class (1 point).

(60) The non–zero fields E⃗ and B⃗ are non–parallel in inertial frame K. Inertial frame K′ moves with velocity v⃗ with respect to K. Find a physical velocity v⃗ so that E⃗′ and B⃗′ are parallel. (Hint: Try E⃗ = E_{2} ê_{2}, B⃗ = B_{2} ê_{2} + B_{3} ê_{3} and v⃗ = v ê_{1}.) Due April 17 before class (10 points).

(61) Transform

f^{i} = \frac{q}{c} F^{iβ} U_{β}

into

f⃗ = q γ E⃗ + \frac{q}{c} U⃗ × B⃗ .

Due April 17 in class (4 points).